We show that the low-energy states of non-Abelian topological orders possess extensive magic which is long-ranged, and cannot be eliminated by a constant-depth local unitary circuit. This refines conventional notions of complexity beyond the linear circuit depth which is required to prepare any topological phase, and provides a new resource-theoretic characterization of topological orders. A central technical result is a no-go theorem establishing that stabilizer states--even up to constant-depth local unitarie--cannot approximate low-energy states of non-Abelian string-net models which satisfy the entanglement bootstrap axioms. Moreover, we show that stabilizer-realizable Abelian string-net phases have mutual braiding phases quantized by the on-site qudit dimension, and that any violation of this condition necessarily implies extensive long-range magic. Extending to higher spatial dimensions, we argue that any state obeying an entanglement area law and hosting excitations with nontrivial fusion spaces must exhibit extensive long-range magic. This applies, in particular, to ground-states and low-energy states of higher-dimensional quantum double models.

翻译：我们证明非阿贝尔拓扑序的低能态具有无法被常数深度局域幺正电路消除的广域长程魔法。这修正了超越制备任意拓扑相所需线性电路深度的复杂性传统概念，并为拓扑序提供了新的资源理论表征。核心技术成果是一个否定性定理：满足纠缠自举公理的非阿贝尔弦网模型的低能态——即使考虑常数深度局域幺正变换——无法用稳定子态近似。此外，我们证明稳定子可实现的阿贝尔弦网相的互编织相位由局域量子位维度量化，任何对该条件的违反必然意味着广域长程魔法。推广到更高空间维度，我们论证任何满足纠缠面积定律且承载具有非平凡融合空间激发的态必然展现出广域长程魔法。这一结论特别适用于高维量子对偶模型的基态和低能态。